Green Tea Sludge?

So I've been trying to explore more of the Japanese teas out there, however I keep running into a recurring problem--namely, the green tea-sludge left at the bottom of my pot when I'm done. I don't know, maybe it's just me, but I find it very obnoxious (and more than a little gross) to clean out--it doesn't come out neatly like whole leaves do, and since I know better than to dump the stuff in my garbage disposal, I end up having to scoop out what I can of the green sludgy mess out of the pot with my fingers before rinsing out the rest, which is not exactly a fun process.

So maybe this is a bit of a silly question, but--am I doing this completely wrong? Is there any easier/better way to do this? Some tool or technique I should use? I do I just need to resign myself to green sludge being the price I pay for a tea-drinking session?

If you don't want to touch the leaves, you could just overfill the pot with water and dump everything into a big container/bowl, a couple of rinses should do the trick, then take the container outside and dump it into nature

I use a small wooden spoon to scoop out most of the leaves. Then I rinse under the tap until the rest is gone. Some sticky ones may have to be removed with a toothbrush. When I'm done I do a final rinse with boiling water (hot pot drys faster).

Thanks for all the ideas--I definitely hadn't thought of the bowl of water/strainer idea! Though I wonder if it's possible to get a biodegradable/natural strainer of some kind, so that I won't feel guilty about throwing yet another bit of plastic into the landfill every year or so? A wicker basket or something, maybe? Hmm ... Sadly, composting is tricky since I live in a condo complex .... I have to be sneaky about it!

Stentor, thank you for the wooden spoon suggestion--I don't have anything that small, but it's definitely something I could pick up!